Uniform Approximation and Bracketing Properties of VC classes

We show that the sets in a family with finite VC dimension can be uniformly approximated within a given error by a finite partition. Immediate corollaries include the fact that VC classes have finite bracketing numbers, satisfy uniform laws of averages under strong dependence, and exhibit uniform mixing. Our results are based on recent work concerning uniform laws of averages for VC classes under ergodic sampling.

💡 Research Summary

The paper investigates fundamental approximation and entropy properties of classes of sets (or indicator functions) that have a finite Vapnik‑Chervonenkis (VC) dimension. The central result is a uniform approximation theorem: for any error tolerance ε > 0 and any class 𝔽 with VC dimension d, one can construct a finite measurable partition Πε of the underlying sample space such that, on each cell of the partition, every set in 𝔽 behaves almost identically—more precisely, the difference between the indicator functions of any two sets in 𝔽 is bounded by ε in L¹ on each cell. The size of the partition grows only polynomially in 1/ε, with an exponent that depends on d, and crucially does not depend on the sample size.

From this uniform approximation theorem the authors derive two immediate corollaries. First, the bracketing number of 𝔽 with respect to the L¹(P) metric is finite for every ε. By taking the lower and upper brackets on each cell of Πε, one obtains a collection of ε‑brackets whose cardinality is bounded by the number of cells, i.e., O((1/ε)^{d}). Hence the bracketing entropy of a VC class is controlled in the same way as its covering entropy, extending classical results that were previously proved only for independent and identically distributed (i.i.d.) samples.

Second, the uniform approximation yields a uniform law of large numbers (ULLN) for 𝔽 under very general dependence structures. The authors build on recent work that established ULLN for VC classes under ergodic sampling. By coupling the empirical averages of the indicator functions with the cell‑wise averages defined by Πε, and using the Birkhoff ergodic theorem together with martingale difference bounds, they show that \